### Math-3510-410 Junior Honors Seminar

Dr. Sarah J. Greenwald

9:00-9:50 a.m. MWF, 314 Walker

While geometry means measuring the earth, too often it is presented in an
axiomatic way, divorced from reality and experiences. But in this course, we
will use intuition from your experiences with hands on models and data to
understand real-world applications of geometry such as the geometry of the
universe and applications of geometry to art, mapping the brain, robotics,
graphics, space shuttle navigation and more! Topics will be chosen by student
interest (see http://www.cs.appstate.edu/~sjg/class/geo.html for more info).
In the process, we'll see the interplay between geometry and linear algebra,
modern algebra, analysis, differential equations, probability and statistics.

**
Prerequisites**: The following
is required:
A 3.0 GPA in math courses, honors status, or permission of the instructor,
and

**any ONE
of the following courses**:
Linear Algebra (MAT 2240),
Sophomore Honors Seminar (MAT 2510),
Modern Algebra (MAT 3110),
Real Variables (MAT 3220), or
Junior Honors Seminar (MAT 3510).

No background in
geometry is needed.

Applications of Geometry - A Small Sampling of Possible
Topics

**Geometry of the Earth** Geometry means measuring the earth.
There are many interesting applications related to the geometry of the earth,
including gps technology.

**Geometry of the Universe**
While the ancient Greeks knew that the earth was round, the debate about the
shape of the earth lasted a long time. There is a similar debate
today about the shape of the universe. Researchers are trying to
find out if our universe has more than 3 spatial dimensions and
if it satisfies the laws of spherical, hyperbolic of Euclidean geometry.
On June 30th, MAP (Microwave Anistropy Probe) was launched.
See some of the following links to see how this is related to the
shape of space.
MAP Status,
Henderson's the Shape of Space,
MAP Mission Goal,
Discover's
The Magnificent Mission

**Rotations and the Space Shuttle**
If you've taken linear algebra, then perhaps you learned about
3d rotation matrices (6.5 of book). To pilot the space shuttle,
computer programs need to know how to rotate. The problem with
using the matrices is that sometimes a phenomenon occurs that is known
as Gimbal Lock. As represented in the movie Apollo 13, they had
problems with this. We'll look at this phenomenon, and today's
solution which involves something called quaternions.

**Shortest or Best Path** This idea is important in geometry
and real life. For example, in robotics, a robot that is going to
perform neurosurgery might not want the shortest path since they
might need to avoid certain areas of the brain. And in
taxi-cab geometry, where streets are set up on a grid and there aren't
any diagonal paths, measurement is different than in Euclidean geometry.
This has applications to ecology - ecological distance between species.

**Geometric Probability** Use pictures and area to help solve
questions about the probability of an event occurring.

**Statistical Symmetry** Use a tiling of the plane called the Penrose
tiling to count the number of times a pattern appears in a circle of
radius r. This is used to model quasicrystals and construct new forms
of graph paper.

**Symmetry, Patterns and Tesselations** These are found
all over the place and have many applications.

**Number Theory** Graphing an equation often leads to new
insights and solutions.

**Art** Perspective drawings (projective geometry)
began in Florence in the 1400s. Many artists still follow these
mathematical rules today.
Picasso and others have tried to depict the 4th spatial dimension
by visually representing the mathematics involved.

**Architecture**
What is the best way to enclose a given
amount of space with the least surface area? How about
2 adjoining rooms. If you have a conjecture, how would you prove it?
The double bubble conjecture was recently proven in 2000, with the
help of some undergraduate students.

**Biology and Chemistry** The structures of DNA, crystals,
and viruses all involve geometry.

**Engineering** How would you build a curved surface, say
a horse saddle, out of wood? If it is a ruled surface, then it is a
piece of cake.

**Physics** Einstein's theory of relativity depends on
Riemannian geometry.

**Medical Imaging** A CAT scan takes 2-d picture slices.
Doctors use the geometry of recovering a surface
from such slices to understand the 3d shape.

**Mapping the Brain**
Road Map for the Mind -
OLD MATHEMATICAL THEOREMS UNFOLD THE HUMAN BRAIN

**Can You Hear the Shape of a Drum?**
If you close your eyes and listen to 2 drums, can you hear a difference
in their shape?

**Cartoon and Movie Special Effects** Notice how cartoon and
movie effects have become much more realistic?
Realistic texture and other properties
are now possible thanks to geometry.